INACCURACIES OF MEASURING METHODS AND THEIR INFLUENCE ON THE REGRESSION FUNCTIONt

The quality parameters of electronic components and devices usually depend on the parameters of the materials. In many cases one does not knowthe theoretical relationship between the parameters, and therefore one makes technological experiments and measures the values of the parameters. Usually it is necessary to take several measuring points and calculate from this the unknown relationship between the parameters. The simplest equation that one can use is the linear function. In this case the theoretical probability density function is a Gaussian-function. Otherwise it is necessary to assume that the linear function is an approximation. When the measuring process has an inaccuracy, then one can showthatthe increase ofthe linear function is smaller and it is necessary to estimate a factor of correction to calculate the theoretical or exact relationship.


INACCURACIES OF MEASURING METHODS
The quality parameters of electronic components and devices usually depend on the parameters of the materials. In many cases one does not know the theoretical relationship between the parameters, and therefore one makes technological experiments and measures the values ofthe parameters. Usually it is necessary to take several measuring points and calculate from this the unknown relationship between the parameters. The simplest equation that one can use is the linear function. In this case the theoretical probability density function is a Gaussian-function. Otherwise it is necessary to assume that the linear function is an approximation. When the measuring process has an inaccuracy, then one can show that the increase ofthe linear function is smaller and it is necessary to estimate a factor of correction to calculate the theoretical or exact relationship.

THE REGRESSION FUNCTION
The quality characteristics of electronic components are largely dependent on the material characteristics, among which also the geometric dimensions will be classified by us, in general show variations in their values, we observe also a variation in the values of the quality characteristics.
Normally, between the different parameters and characteristics there exist relations of a physical or chemical type, which are not precisely known, due to a large number of influencing factors. For instance, it has been estimated that the yield in the manufacturing of integrated circuits is influenced by 500-600 factors.
In production engineering the required relationships are determined empirically. Let us consider here an influencing quantity X and a quality characteristic Y. In the Figure 1, as an example, the reverse voltage is represented as a function of the temperature during rinsing. The points are determined by measuring. The science of engineering now requires methods for constructing a relation from this point distribution.
One possibility consists in determining the relative number of points in a unit area and estimating from it the so-called density function f(x, y). Of course, this is very timeconsuming, but it would reflect the relationship quite well.
Since with two-dimensional functions the relation between X and Y is not obtained directly, one calculates with the so-called regression function.
It represents the conditional expectation value The expression in the denominator is called boundary density fy(X): fy(X)-ff(x,y) dy. (2) Paper originally presented at the 5th International Spring Seminar on Electrotechnology held at Prenet, Czechoslovakia, 1-4 June, 1982.

THE INACCURACY OF THE MEASURING PROCESS
By the measuring process the value u is determined for the true value x, and the value v for the true value y. Of course, x and u or y and v may agree, but they differ from each other, in general.
Here, also, a two-dimensional density g(u, v; x, y) is introduced, which, however, is still dependent on x and y. In many cases holds g(u, v; x, y) g(u-x, v-y) g(x', y'), i.e., density depends only on the difference between the read value and the true one. By this, the density can be interpreted as a two-dimensional density for the measuring process as well as for the production engineering process, and the same considerations hold with respect to a regression function.
For instance, for the "pure" measuring process alone there can be given a regression line (x) where s xy is the covariance of the measuring processes and Sx is the variance of the boundary distribution, x' and y' are the systematic measuring errors with respect to x and y (expectation values of the boundary distributions). Frequently, the measurements for x and y are stochastically independent, i.e., S'xy 0.
Instead of the variables x' and y', in mathematical operations frequently x and y are also usedthis being only a question of designation.

THE PRODUCTION ENGINEERING PROCESS AND THE INACCURACY OF MEASURING
Since the measuring process does not have any influence on the stochastic properties of the production engineering processes, and vice versa, the total process is determined by convolution: h(x, y) f(x, y) * g(x, y) +oo f(x', y') g(x-x', y-y') dx' dy'.
Then, if the parameters of the total process are labelled with an asterisk (*)and the parameters of the measuring process with a prime, one obtains: is determined by measurement and the relationship ofthe pure production engineering process is calculated from it, the stochastic measuring error Sx 'z being known: (Sx .2 --Sx)" y --Sy) Thus, by the measurement also the correlation coefficient and thus the degree of determinacy a is reduced. Of course, the test for stochastic independence must be carried out with r and not with r*.

GENERAL RELATIONSHIPS
Also, with nonlinear regression functions the influence ofthe measuring process can be predicted. For this, one requires the boundary density gy(X) ofthe measuring process, or the measuring density g(x) in the case of a stochastic independence. Then, [/dy(X)'fy(X)l*gy(X) /.t(x) fy(X)* gy (x) Of course, this expression is not easy to handle.

APPLICATIONS
For the first time, these methods were used to analyse production engineering processes of microelectronics. Here, however, the general validity existing for any measuring process should be pointed out, also when the inaccuracy ofmeasuring may be neglected now and then.